| Title:
|
A new approach for solving nonlinear BVP's on the half-line for second order equations and applications (English) |
| Author:
|
Matucci, Serena |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
140 |
| Issue:
|
2 |
| Year:
|
2015 |
| Pages:
|
153-169 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We present a new approach to solving boundary value problems on noncompact intervals for second order differential equations in case of nonlocal conditions. Then we apply it to some problems in which an initial condition, an asymptotic condition and a global condition is present. The abstract method is based on the solvability of two auxiliary boundary value problems on compact and on noncompact intervals, and uses some continuity arguments and analysis in the phase space. As shown in the applications, Kneser-type properties of solutions on compact intervals and a priori bounds of solutions on noncompact intervals are key ingredients for the solvability of the problems considered, as well as the properties of principal solutions of an associated half-linear equation. The application of this method leads to some new existence results, which complement and extend some previous ones in the literature. (English) |
| Keyword:
|
global solution |
| Keyword:
|
nonlocal boundary value problem |
| Keyword:
|
noncompact interval |
| Keyword:
|
continuous dependence of solution |
| Keyword:
|
fixed point theorem |
| Keyword:
|
principal solution |
| MSC:
|
34A40 |
| MSC:
|
34B10 |
| MSC:
|
34B15 |
| MSC:
|
34B18 |
| MSC:
|
34B40 |
| idZBL:
|
Zbl 06486931 |
| idMR:
|
MR3368491 |
| DOI:
|
10.21136/MB.2015.144323 |
| . |
| Date available:
|
2015-06-30T12:15:38Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144323 |
| . |
| Reference:
|
[1] Andres, J., Górniewicz, L.: Topological Fixed Point Principles for Boundary Value Problems.Topological Fixed Point Theory and Its Applications 1 Kluwer Academic Publishers, Dordrecht (2003). Zbl 1029.55002, MR 1998968 |
| Reference:
|
[2] Cecchi, M., Došlá, Z., Marini, M.: Principal solutions and minimal sets of quasilinear differential equations.Dyn. Syst. Appl. 13 (2004), 221-232. Zbl 1123.34026, MR 2140874 |
| Reference:
|
[3] Chanturiya, T. A.: Monotone solutions of a system of nonlinear differential equations.Ann. Pol. Math. 37 Russian (1980), 59-70. MR 0574996 |
| Reference:
|
[4] Coppel, W. A.: Stability and Asymptotic Behavior of Differential Equations.D. C. Heath and Company VIII Boston (1965). Zbl 0154.09301, MR 0190463 |
| Reference:
|
[5] Došlá, Z., Marini, M., Matucci, S.: Positive solutions of nonlocal continuous second order BVP's.Dyn. Syst. Appl. 23 (2014), 431-446. Zbl 1312.34057, MR 3241888 |
| Reference:
|
[6] Došlá, Z., Marini, M., Matucci, S.: On some boundary value problems for second order nonlinear differential equations.Math. Bohem. 137 (2012), 113-122. Zbl 1265.34113, MR 2978257 |
| Reference:
|
[7] Došlá, Z., Marini, M., Matucci, S.: A boundary value problem on a half-line for differential equations with indefinite weight.Commun. Appl. Anal. 15 (2011), 341-352. Zbl 1244.34045, MR 2867356 |
| Reference:
|
[8] Došlý, O., Řehák, P.: Half-Linear Differential Equations.North-Holland Mathematics Studies 202 Elsevier, Amsterdam (2005). Zbl 1090.34001, MR 2158903 |
| Reference:
|
[9] Elbert, Á., Kusano, T.: Principal solutions of non-oscillatory half-linear differential equations.Adv. Math. Sci. Appl. 8 (1998), 745-759. Zbl 0914.34031, MR 1657164 |
| Reference:
|
[10] Erbe, L. H., Wang, H.: On the existence of positive solutions of ordinary differential equations.Proc. Am. Math. Soc. 120 (1994), 743-748. Zbl 0802.34018, MR 1204373, 10.1090/S0002-9939-1994-1204373-9 |
| Reference:
|
[11] Franco, D., Infante, G., Perán, J.: A new criterion for the existence of multiple solutions in cones.Proc. R. Soc. Edinb., Sect. A, Math. 142 (2012), 1043-1050. Zbl 1264.47059, MR 2981023, 10.1017/S0308210511001016 |
| Reference:
|
[12] Gaudenzi, M., Habets, P., Zanolin, F.: An example of a superlinear problem with multiple positive solutions.Atti Semin. Mat. Fis. Univ. Modena 51 (2003), 259-272. Zbl 1221.34057, MR 2045073 |
| Reference:
|
[13] Hartman, P.: Ordinary Differential Equations.Birkhäuser, Boston (1982). Zbl 0476.34002, MR 0658490 |
| Reference:
|
[14] Kiguradze, I., Půža, B.: Boundary Value Problems for Systems of Linear Functional Differential Equations.Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math. 12 Masaryk University, Brno (2003). Zbl 1161.34300, MR 2001509 |
| Reference:
|
[15] Marini, M., Matucci, S.: A boundary value problem on the half-line for superlinear differential equations with changing sign weight.Rend. Ist. Mat. Univ. Trieste 44 (2012), 117-132. Zbl 1269.34027, MR 3019556 |
| Reference:
|
[16] Mirzov, J. D.: Asymptotic Properties of Solutions of Systems of Nonlinear Nonautonomous Ordinary Differential Equations.Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math. 14 Masaryk University, Brno (2004). Zbl 1154.34300, MR 2144761 |
| Reference:
|
[17] Motreanu, D., Rădulescu, V.: Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems.Nonconvex Optimization and Its Applications 67 Kluwer Academic Publishers, Dordrecht (2003). Zbl 1040.49001, MR 1985870, 10.1007/978-1-4757-6921-0_5 |
| Reference:
|
[18] Powers, D. L.: Boundary Value Problems and Partial Differential Equations.Elsevier/Academic Press, Amsterdam (2010). Zbl 1187.35001, MR 2584516 |
| Reference:
|
[19] Wang, J.: The existence of positive solutions for the one-dimensional $p$-Laplacian.Proc. Am. Math. Soc. 125 (1997), 2275-2283. Zbl 0884.34032, MR 1423340, 10.1090/S0002-9939-97-04148-8 |
| . |
